Table Of ContentPreferential attachment scale-free growth model
with random fitness
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0 Marcelo D. S. de Meneses 1 , Sharon D. da Cunha 1 ,
2
D. J. B. Soares 1,2 and L. R. da Silva 1
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a
J
2 1Departamento de F´ısica Teo´rica e Experimental,
1
Universidade Federal do Rio Grande do Norte,
Campus Universitario, 59072-970 Natal-RN, Brazil
]
h 2Departamento de Biof´isica e Farmacologia, Centro de Biociˆencias,
c
Universidade Federal do Rio Grande do Norte,
e
m Campus Universita´rio 59072 970, Natal-RN, Brazil
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t
a
Weintroduceamodelwhichconsistsinaplanarnetworkwhichgrows
t
s by adding nodes at a distance r from the pre-existing barycenter. Each
.
t newnodepositionisrandomlylocatedthroughthedistributionlawP(r)∝
a 1/rγ with γ >1. The new node j is linked to only one pre-existing node
m
accordingtotheprobabilitylawP(i↔j)∝η k /rαA (1≤i<j;α ≥0);
i i ij A
- k isthenumberoflinksoftheithnode,η isitsfitness(orqualityfactor),
d i i
n andrijisthedistance. Weconsiderinthepresentpapertwomodelsforηi.
o In one of them, the single fitness model (SFM[1]), we consider ηi =1∀i.
c In the other one, the uniformly distributed fitness model (UDFM), ηi is
[ chosentobeuniformlydistributedwithintheinterval(0,1]. Wehavede-
termined numerically thedegree distribution P(k). This distribution ap-
1
v pearsto bewell fittedwith P(k)=P(1)k−λe−q(k−1)/κ with q≥1, where
3 exq ≡[1+(1−q)x]1/(1−q) (ex1 =ex)istheq-exponentialfunctionnaturally
7 emerging within nonextensive statistical mechanics. We determine, for
2 both models, the entropic index q as a function of α (q independs from
A
1 γ). Additionally, we determine theaverage topological (or chemical) dis-
0 tance within the network, and the time evolution of the average number
6 of links hk i. We obtain that, asymptotically, hk i ∝ (t/i)β, (i coincides
i i
0
with the input-timeof thei node) and β(α ) to both cases.
/ A
t
a Scale-free networks are very popular nowadays [2, 3, 4, 5] due to their un-
m countable applications in different fields of knowledge. These models are typi-
- cally associated with some physical quantities that are characterizedby power-
d
law asymptotic behavior, instead of the usual exponential laws. Most of these
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models do not take into account the node-to-node Euclidean distance, i.e., the
o
c geographicaldistance. One of them whichdoes take into accountthis aspectof
: the problem has been introduced recently [1] and discussed. In this example,
v
i as well as in others [6, 7], strong connection has been revealed with nonexten-
X
sive statistical mechanics [8, 9]. In the present paper, we follow the lines of [1]
r which we extend in the sense that we include now a local variabledenominated
a
fitness or quality factor, we call this variable η . The model studied by [1] is
i
hereinreferredto asthe Single Fitness Model (SFM), thatis, η =1∀i,andwe
i
1
alsostudytheUniformly DistributedFitness Model(UDFM),inwhichthelocal
fitness is an independent random variable. We are here interested to focus on
the effects ofthe fitness onthe connectivityprobabilitydistributionandsimilar
quantities.
Figure1: ConectivitydistributionP(k)inlog-logscalefortypicalvaluesα for
A
UDFM and SFM models. The symbols are numerical results and continuous
lines are the best fits in according to equation 2.
Figure 2: Representation of the figure 1 in ln - linear.
q
At the plane, we have constructed a complex network which is constituted
by nodes and links between nodes. The network grows sequentially by adding
a node ateachtime. We fix the firstnode (i=1)atthe origin. The positionof
the second node (i = 2) is chosen randomly with respect to origin in such way
2
its distance r obeys the probability law P(r)∝1/rγ with γ >1. In this paper,
we have chosen γ = 5/4 for all cases studied. Next the second node is linked
to first one. Then, the origin is moved into network barycenter formed by the
twonodes. The previousprocedureis usedto include the third site,fourthsite,
and so on. We link the third node (j = 3) to one of the pre-existing nodes in
the network. We establish its connection by considering the linking probability
given by P(i ↔ j = 3) ∝ η k /rαA (α ≥ 0) and 1 ≤ i < j. The connectivity
i i ij A
ofith node isstoodfork andη isits fitness andr is its distanceto newnode
i i ij
(j =3). At this early stage we have k =1 and k =1.
1 2
Figure 3: α -dependence of (q, λ, κ) for both SFM and UDFM models.
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The Fitness model is characterized by the presence of factor η that appear
i
in the linking probability. We define the fitness or quality model in following
way: To each node i < j is assigned a uniform random value η ∈ (0,1] when
i
the node is born. That value is kept constant while the network grows. In
traditional models old nodes are more attractive than new ones. In this model
we give a chance for new sites, with high fitness, become competitive.
The network growth process is sequentially repeated up to size desired for
network. Toavoidnodepositionoverlapweconsiderthelowestdistancebetween
themtobeunit. IfwedenoteN thetotalnumberofnodesonecanwritelinking
probability for ith network node to connect the new node j =N +1 as
η k /rαA
P(i↔j =N +1)= i i i (1)
N η k /rαA
i=1 i i ij
P
The dynamics that we have introduced by the above rule does privilege the
connectionbetweennewnodesandthosehavingmanylinks(highfitnessnodes)
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since they arenotsofarfromeachother. Onesimilarrule wasexplored[10]for
theparticularcaseofuniformdistributionofsiteswithinsomelimitedregionat
the plane, with η =1∀i. The α parameter control the influence range of the
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nodes (or hubs). The case α zero corresponds to the Bianconi fitness model
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[11] where distance is not relevant. In the present paper we will emphasize the
following aspects:
• The network degree distribution in the stationary-state P(k) relative to
number of nodes having k links in the N →∞.
• The temporal dependence of average number of links hk i and more pre-
i
cisely to know how asymptotically it increase with time t/i, t≥i.
• The dependence of average length, hli, with α parameter.
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Thenumericalresultsareobtainedfornetworkswith10000nodesand2000
runs. In this network size the probability distribution P(k) practically enter
in the stationary regime. The figure 1 show our numerical results for P(k)
(symbols) and fits (continuous lines) for UDFM and SFM models. One can
observe in our results that α has a strong influence on P(k). The present
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results are very well fitted by the function
P(k)=P(1)k−λe−(k−1)/κ (2)
q
where q-exponential function is defined by
ex ≡[1+(1−q)x]1/(1−q) (ex =ex). (3)
q 1
The variable κ is the characteristic number of links [1]. In the figure 2 we
representthe figure 1 in a ln - linear plot. Where ln (x)=(x1−q−1)/(1−q).
q q
Figure 4: Temporal dependency of the average connectivity for UDFM model.
Analizingthefigure1,wehavefoundthattheUDFMhasraisedthenumber
of links per node and it turns out that hubs are more connected. In the figure
3we presentthe (q, λ, κ)evolutioncurvewithα forbothmodels studied. We
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observe that UDFM curve is always above the SFM case. They have the same
asymptotic limit, i.e., q(α )=1.
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Figure 5: Average connectivity exponent for α values relative to measures on
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node i=50.
Figure 6: Average path length for UDFM and SFM models.
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Theresultsforthe node averageconnectivityhk iareindicatedinthe figure
i
4 where we show its temporal evolution. Our data for hk i can be fitted by the
i
function
t β(αA,ηi)
hk i∝ t≥i (4)
i
(cid:18)i(cid:19)
Thecollapseofthecurvesforα ≥2indicatesaupperlimit(naturalcutoff)
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for a node connectivity. As much we increase α all hubs reduce its connec-
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tivity to a minimal value 2. We explain this result imagining that the linking
probability decays strongly for increasing α in such way that a node does not
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see other unless its first neighbours, and therefore they form long node chains.
This fact is inherent to the network growing rule, i.e., if we use other rule for
example each new site links with more than one node, this limit result would
be quite different.
In the figure 5 we show the exponent β(α ) for UDFM and SFM models.
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The exponent decline down from 0.59 for UDFM and from 0.48 for SFM to a
stationary value 0.1. The figure 6 shows the average length hli dependency on
α for UDFM and SFM models. The average path length, hli, of a network is
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definedasthenumberofedgesintheshortestpathbetweentwonodesaveraged
over all pairs of nodes. We see that, when α varies, hli raises rapidly to
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a saturated value ∼ 12. The saturation in figure 6 is explained by the hk i
i
convergence in figures 5 and 4 for α ≥2.
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Now we will summarize our present paper. Following the lines of the paper
[1] we have presented a model in which we add fitness. We study the effect of
competition between the relevant variable for connectedness when we include
fitness on the network. In UDFM model the popular nodes (many links) do
compete with younger nodes when the fitness is a important factor that per-
mits them to get more links. When we raise α we favor the linking between
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first neighbours in the network. When α is zero we recover the Bianconi-
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Baraba´simodel [11]. The averageconnectivity hk i and averagepath length hli
i
are appreciably influenced by factor α for UDFM and SFM models. We have
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shown that degree distribution P(k) is very well fitted by the funtion proposed
that contain q-exponential function that emerges naturally from non-extensive
statistics. And when we compare our results relative to SFM we have obtained
that the number of links per node has raised for all values α . When the α is
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increasedwefindthatlinksperhubs hasreducedasSFMbutitoccoursslower.
The presence of fitness changes the universality class in the complex network.
Thenifyouwantamorecompletescenarioforthiskindofsystem,fitnessshould
be included.
Acknowledgments
We acknowledge Constantino Tsallis for critical reading and many fruitful sug-
gestions, as well as financial support from Capes, Pronex/MCT and CNPq
(Brazilian agencies). One of us (LRS) also acknowledges warm hospitality at
the Santa Fe Institute, where this work was completed.
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References
[1] D.J.B. Soares, C. Tsallis, A.M. Mariz and L.R. da Silva, Europhys.
Lett.,70,(2005),70.
[2] D.J. Watts and S.H. Strogatz, Nature,393, (1998), 440.
[3] R. Albert and A.-L. Barabasi,Science ,286, (1999), 509.
[4] R. Albert and A.-L. Barabasi,Rev. Mod. Phys.,74, (2002), 47.
[5] A.F.Rozenfeld,R.Cohen,D.ben-AvrahamandS.Havlin,Phys.Rev.Lett.
89,(2002),218701.
[6] S. Thurner and C. Tsallis, Europhys. Lett. 72, 197 (2005).
[7] D.R. White, N. Kejzar, C. Tsallis, D. Farmer and S. White,
cond-mat/0508028.
[8] C. Tsallis, J. Stat. Phys. 52, 479 (1988). For full bibliography see
http://tsallis.cat.cbpf.br/biblio.htm
[9] M.Gell-MannandC.Tsallis,eds.,NonextensiveEntropy-Interdisciplinary
Applications (Oxford University Press, New York, 2004).
[10] S.S. Manna and P. Sen, Phys. Rev. Lett.66 ,(2002), 066114.
[11] G. Bianconi and A.-L. Barabasi, Europhys. Lett., 54,(2001),436.
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